Does 10 have a friend?

Everybody has friends, that's a fact.

But do you know numbers have also friends. Though they share a special kind of friendship in which they only become friendly if they fulfil a certain condition.

That certain condition is called the abundancy(σ(n)). If a number share the same abundancy with another number, then it is called a 'friendly' number. And if not, it is called a 'solitary' number.

The abundancy is simply the ratio of sum of divisors of the number and the number itself.

To understand it let's take an example. Consider the number 6.

It's divisors are 1, 2, 3 and 6. Their sum is 12.

Now, the abundancy of 6, σ(6) = 12/6 = 2.

6 is also the smallest friendly number which shares its abundancy with number 28 which is 2.

If you thought for a second, then you will realize that all prime numbers are solitary numbers. You can prove it by yourself. (Hint: The denominator of the abundance will be a prime number.)

There are more solitary numbers than friendly numbers. Really, numbers are also introverts these days.

You know, no one knows whether 10 is a solitary number Or not.

It's abundacy is 9/5. No number is found to be with same abundancy, but it is not proved that it is a solitary number.